Editing Order (ring theory) (section)

==Algebraic number theory==
The leading example is the case where ''<math>A</math>'' is a [[number field]] ''<math>K</math>'' and <math>\mathcal{O}</math> is its [[ring of integers]]. In [[algebraic number theory]] there are examples for any ''<math>K</math>'' other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the [[field extension]] ''<math>A=\mathbb{Q}(i)</math>'' of [[Gaussian rational]]s over <math>\mathbb{Q}</math>, the integral closure of ''<math>\mathbb{Z}</math>''  is the ring of [[Gaussian integer]]s ''<math>\mathbb{Z}[i]</math>'' and so this is the unique ''maximal'' ''<math>\mathbb{Z}</math>''-order: all other orders in ''<math>A</math>'' are contained in it. For example, we can take the subring of [[complex number]]s of the form <math>a+2bi</math>, with <math>a</math> and <math>b</math> integers.<ref>Pohst and Zassenhaus (1989) p.&nbsp;22</ref>

The maximal order question can be examined at a [[local field]] level. This technique is applied in algebraic number theory and [[modular representation theory]].